Lindenmayer Systems and Turtle Graphics

Mr. Worley

2025 Oct 06

Lindenmayer Systems

B   B   - B   B   -- B   B   - B   A    C   - C   A    -- C   A    - B   A    C   ---
BB- BB- - BB- BB- -- BB- BB- - BB- BAC- CA- - CA- BAC- -- CA- BAC- - BB- BAC- CA- ---

Turtle Graphics

Imagine a turtle dragging a pen across an \((x,y)\) plane. If various characters (letters) in a string represent commands, we can see that L systems lead to fractal-like patterns.

A common set of commands:

So, using our example from the previous page, and starting pointing North (up)…

BB-BB--BB-BB---BB-BB--BB-BAC-CA--CA-BAC---CA-BAC--BB-BAC-CA----

Example 1

Angle = 30°
Start = A
Rules:
  A > A[+A][-A]B
  B > [--B]A
Number of recursions = 7

Example 2

Angle = 60°
Start = A
Rules:
  A > -B+B+B+B+B+B
  B > B[--CCA][-CCA]B
  C > CC
Number of recursions = 7

Example 3

Angle = 45°
Start = A
Rules:
  A > [----AAB][--AAB][++AAB]AAB
  B > [-C][+C][--C][++C][---C][+++C]
  C > CAB
Number of recursions = 6

Example 4

Angle = 72°
Start = A
Rules:
  A > [-AAB][--AAB][+AAB][++AAB]AAB
  B > [-C][+C][--C][++C]CA
  C > CC
Number of recursions = 5

Example 5

Angle = 60°
Start = a
Rules:
  a > -B+B+B+B+B+B
  B > BB[-CCa]
  C > CC
Number of recursions = 9

Example 6

Angle = 72°
Start = A
Rules:
  A > B+B+B+B+B
  B > [-CCA]BB
  C > CC
Number of recursions = 9

Example 7

Angle = 45°
Start = A
Rules:
  A > +A-B-B[B]
  B > -B+A+A[A]
Number of recursions = 8

Example 8

Angle = 90°
Start = A
Rules:
  A > AA[-A][+A]
Number of recursions = 8

Example 9

Angle = 30°
Start = A
Rules:
  A > AcbcA
  b > [+A][-A][++A][--A][+++A][---A][++++A][----A][+++++A][-----A]
  c > bAb
Number of recursions = 6

Example 10

Angle = 30°
Start = AB
Rules:
  A > B--B
  B > AA+++AA
Number of recursions = 12

Example 11

Angle = 30°
Start = x
Rules:
  x > FFF[+FFF+FFF+FF+FF+F+Fx][-FFF-FFF-FF-FF-F-Fx]Fx
  F > FFx
Number of recursions = 6

Example 12

Angle = 45°
Start = A
Rules:
  A > AB
  B > A[++C][--C]
  C > C[+C][-C]A
Number of recursions = 10

Example 13

Angle = 25°
Start = -x
Rules:
  x > F+[[x]-x]-F[-Fx]+x
  F > FF
Number of recursions = 7

Example 14

Angle = 90°
Start = b
Rules:
  A > AA
  b > AbA+AbA+AbA
Number of recursions = 9

Example 15

Angle = 115°
Start = A
Rules:
  A > A+A-A
Number of recursions = 11

Example 16

Angle = 90°
Start = A
Rules:
  A > A[+AA][-AA]B
  B > ABA
Number of recursions = 7

Example 17

Angle = 6°
Start = x
Rules:
  x > FFF[+FF+FF+FF+F+Fx][-FF-FF-FF-F-Fx]FFFFFFFFFFFFFFx
  F > FF
Number of recursions = 8